94 Mr. J. Brill on "Densities hi the Earth's Crust" 



We will change the variables in this expression by writing 

 cos 6 = /j, and r = ah. The expression then becomes 



A 2 (l —hfi) , 7 , . 



ap (l-2M+^ ***' 



where we have rejected the negative sign that arises in the 

 transformation from 6 to //-, as we afterwards intend to 

 integrate with regard to /j, in the direction in which that 

 variable increases. 



Now, if h be less than unity, as it is in the case we are con- 

 sidering, we may write 



and therefore 



(1 _^^ / ^ i =/^ + /K^Pl~l)+...+/t W (/^Pn--Pn--l)+<fec. 



Now, we have the formula 

 and therefore 



h^n — P n — 1= 7j — ~T—T (P>i + 1 — Pn-l). 



Thus we obtain 



fi — h „ , ,^ „, n + 1 





( i_v.+^ = ' i+§A(P, - 1,+ ••• +^n / '"( p - + '- I> -'> +&c - 



If we differentiate this with respect to /jl, we deduce 



(1-2M + A 2 )'" +3 ^ ' + 2n+l" V dfjL dfi r^ C * 



= 1+2M\ + ... +(n + l)A*P. + Ao., 



since 



^Pn+i _ ^-i = (2n + l)P n . 

 dfju djjij 



Thus the expression for the portion contributed to the value 

 of gravity at P by our element is 



apdhd f id<f>{h 2 + 2h 3 P 1 + ... +>-l)/i"P n _ 2 + &c.}. 



Imagine a solid element cut out of the crust by radii drawn 

 from the Earth's centre to all points of the contour of a polar 

 surface-element drawn upon the Earth's surface. The portion 

 contributed to the value of gravity at P by our solid element 



